A conservation approach to helicoidal surfaces of constant mean curvature in R^3, S^3 and H^3

Details A conservation approach to helicoidal surfaces of constant mean curvature in R^3, S^3 and H^3 A conservation approach to helicoidal surfaces of constant mean curvature in R^3, S^3 and H^3

2011-10-05

arxiv.org/abs/1110.1068v1

Mathematics

Differential Geometry

11 pages, product of the Indiana University REU program 2011

Scientific paper

We develop a conservation law for constant mean curvature (CMC) surfaces introduced by Korevaar, Kusner and Solomon, and provide a converse, so as to characterize CMC surfaces by a conservation law. We work with `twizzler' construction, which applies a screw-motion to some base curve. We show that, excluding cylinders, CMC helicoidal surfaces can be completely determined by a first-order ODE of the base curve. Further, we demonstrate that in R^3 this condition is equivalent to the treadmillsled characterization of helicoidal CMC surfaces given by O. Perdomo.

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